Well-posedness and long-time behavior for the Westervelt equation with absorbing boundary conditions of order zero

TL;DR

This paper proves global well-posedness, instant regularization, and exponential stability of solutions for the Westervelt equation with nonlinear absorbing boundary conditions of order zero, advancing understanding in nonlinear acoustics.

Contribution

It establishes the first rigorous analysis of the Westervelt equation with these boundary conditions, including well-posedness, regularization, and stability results.

Findings

Solutions are globally well-posed for small initial data.

Solutions regularize instantaneously to smooth functions.

Solutions near equilibrium converge exponentially to an equilibrium.

Abstract

We investigate the Westervelt equation from nonlinear acoustics, subject to nonlinear absorbing boundary conditions of order zero, which were recently proposed by Kaltenbacher & Shevchenko. We apply the concept of maximal regularity of type $L_p$ to prove global well-posedness for small initial data. Moreover, we show that the solutions regularize instantaneously which means that they are $C^\infty$ with respect to time $t$ as soon as $t>0$. Finally, we show that each equilibrium is stable and each solution which starts sufficiently close to an equilibrium converges at an exponential rate to a possibly different equilibrium.